Optimal. Leaf size=20 \[ -\frac {\text {Li}_2\left (\frac {e x^n}{d}+1\right )}{e n} \]
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Rubi [A] time = 0.07, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2336, 2315} \[ -\frac {\text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{e n} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2336
Rubi steps
\begin {align*} \int \frac {x^{-1+n} \log \left (-\frac {e x^n}{d}\right )}{d+e x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}\\ &=-\frac {\text {Li}_2\left (1+\frac {e x^n}{d}\right )}{e n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.05 \[ -\frac {\text {Li}_2\left (\frac {e x^n+d}{d}\right )}{e n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 55, normalized size = 2.75 \[ \frac {n \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) + \log \left (e x^{n} + d\right ) \log \left (-\frac {e}{d}\right ) + {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right )}{e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{n - 1} \log \left (-\frac {e x^{n}}{d}\right )}{e x^{n} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 19, normalized size = 0.95 \[ -\frac {\dilog \left (-\frac {e \,x^{n}}{d}\right )}{e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.67, size = 64, normalized size = 3.20 \[ -\frac {{\left (\log \relax (d) - \log \relax (e)\right )} \log \left (\frac {e x^{n} + d}{e}\right )}{e n} + \frac {\log \left (\frac {e x^{n}}{d} + 1\right ) \log \left (-x^{n}\right ) + {\rm Li}_2\left (-\frac {e x^{n}}{d}\right )}{e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 18, normalized size = 0.90 \[ -\frac {{\mathrm {Li}}_{\mathrm {2}}\left (-\frac {e\,x^n}{d}\right )}{e\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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